Fourier series, sine series, cosine series Prof. Joyner 1

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1 Fourier series, sine series, cosine series Prof. Joyner History: Fourier series were discovered by J. Fourier, a Frenchman who was a mathematician among other things. In fact, Fourier was Napoleon s scientific advisor during France s invasion of Egypt in the late 8 s. When Napoleon returned to France, he elected (i.e., appointed) Fourier to be a Prefect - basically an important administrative post where he oversaw some large construction projects, such as highway constructions. It was during this time when Fourier worked on the theory of heat on the side. His solution to the heat equation is basically what undergraduates often learn in a DEs with BVPs class. The exception being that our understanding of Fourier series now is much better than what was known in the early 8 s and some of these facts, like Dirichlet s theorem, are covered as well. Motivation: Fourier series, since series, and cosine series are all expansions for a function f(x), much in the same way that a Taylor series a + a (x x ) + a (x x ) +... is an expansion. Both Fourier and Taylor series can be used to approximate f(x). There are at least three important differences between the two types of series. () For a function to have a Taylor series it must be differentiable, whereas for a Fourier series it does not even have to be continuous. () Another difference is that the Taylor series is typically not periodic (though it can be in some cases), whereas a Fourier series is always periodic. () Finally, the Taylor series (when it converges) always converges to the function f(x), but the Fourier series may not (see Dirichlet s theorem below for a more precise description of what happens). Definitions: Let f(x) be a function defined on an interval of the real line. We allow f(x) to be discontinuous but the points in this interval where f(x) is discontinuous must be finite in number and must be jump discontinuities. First, we discuss Fourier series. To have a Fourier series you must be given two things: () a period P = L, () a function f(x) defined on an interval These notes licensed under Attribution-ShareAlike Creative Commons license, Originally written 6. Last modified The graphics were created by the author using GIMP and SAGE. Remember the formula for the n-th Taylor series coefficient centered at x = x - a n = f(n) (x ) n!?

2 of length L, usually we take L < x < L (but sometimes < x < L is used instead). The Fourier series of f(x) with period L is f(x) a + [a n cos( nx L ) + b n sin( nx L )], where a n and b n are given by the formulas, and a n = L b n = L L L L L f(x) cos( nx )dx, () L f(x) sin( nx )dx. () L Next, we discuss cosine series. To have a cosine series you must be given two things: () a period P = L, () a function f(x) defined on the interval of length L, < x < L. The cosine series of f(x) with period L is where a n is given by f(x) a + a n cos( nx L ), a n = L L cos( nx )f(x) dx. L The cosine series of f(x) is exactly the same as the Fourier series of the even extension of f(x), defined by { f(x), < x < L, f even (x) = f( x), L < x <. Finally, we define sine series. To have a sine series you must be given two things: () a period P = L, () a function f(x) defined on the interval of length L, < x < L. The sine series of f(x) with period L is f(x) b n sin( nx L ), These formulas were not known to Fourier. To compute the Fourier coefficients a n,b n he used sometimes ingenious round-about methods using large systems of equations.

3 where b n is given by b n = L L sin( nx )f(x) dx. L The sine series of f(x) is exactly the same as the Fourier series of the odd extension of f(x), defined by { f(x), < x < L, f odd (x) = f( x), L < x <. One last definition: the symbol is used above instead of = because of the fact that was pointed out above: the Fourier series may not converge to f(x). Do you remember right-hand and left-hand limits from calculus? Recall they are denoted f(x+) = lim ǫ,ǫ> f(x+ǫ) and f(x ) = lim ǫ,ǫ> f(x ǫ), resp.. The meaning of is that the series does necessarily not converge to the value of f(x) at every point 4. The convergence properties are given by the theorem below. Dirichlet s theorem 5 : Let f(x) be a function as above and let L < x < L. The Fourier series of f(x), f(x) a + [a n cos( nx L ) + b n sin( nx L )], (where a n and b n are as in the formulas (), ()) converges to f(x+) + f(x ). In other words, the Fourier series of f(x) converges to f(x) only if f(x) is continuous at x. If f(x) is not continuous at x then then Fourier series of f(x) converges to the midpoint of the jump. Examples: () If f(x) = + x, < x <, then the definition of L implies L =. Without even computing the Fourier series, we can evaluate it using Dirichlet s theorem. Question: Using periodicity and Dirichlet s theorem, find the value that the Fourier series of f(x) converges to at x =,,. (Ans: f(x) is continuous at, so the Fourier series at x = converges to f() = by Dirichlet s theorem. 4 Fourier believed his series converged to the function in the early 8 s but we now know this is not always true. 5 Pronounced Dear-ish-lay.

4 f(x) is not defined at. Its Fourier series is periodic with period 4, so at x = the Fourier series converges to f(+)+f( ) = +4 =. f(x) is not defined at. It s Fourier series is periodic with period 4, so at x = the Fourier series converges to f( )+f( +) = + =.) The formulas () and () enable us to compute the Fourier series coefficients a, a n and b n. (We skip the details.) These formulas give that the Fourier series of f(x) is f(x) n cos (n) n The Fourier series approximations to f(x) are sin( nx ). S =, S = + 4 sin(x ), S = + 4 sin ( x) sin ( x),... The graphs of each of these functions get closer and closer to the graph of f(x) on the interval < x <. For instance, the graph of f(x) and of S 8 are given below: Notice that f(x) is only defined from < x < yet the Fourier series is not only defined everywhere but is periodic with period P = L = 4. Also, notice that S 8 is not a bad approximation to f(x). This can also be done in SAGE. First, we define the function. SAGE sage: f = lambda x:x+ sage: f = Piecewise([[(-,),f]]) This can be plotted using the command f.plot().show(). Next, we compute the Fourier series coefficients: SAGE sage: f.fourier_series_cosine_coefficient(,) # a_ 4 sage: f.fourier_series_cosine_coefficient(,) # a_ sage: f.fourier_series_cosine_coefficient(,) # a_ 4

5 4 4 4 x Figure : Graph of f(x) and a Fourier series approximation of f(x). sage: f.fourier_series_cosine_coefficient(,) # a_ sage: f.fourier_series_sine_coefficient(,) # b_ 4/pi sage: f.fourier_series_sine_coefficient(,) # b_ -/pi sage: f.fourier_series_sine_coefficient(,) # b_ 4/(*pi) Finally, the partial Fourier series and it s plot verses the function can be computed using the following SAGE commands. sage: f.fourier_series_partial_sum(,) SAGE 5

6 -*sin(pi*x)/pi + 4*sin(pi*x/)/pi + sage: P = f.plot_fourier_series_partial_sum(5,,-5,5,linestyle=":") sage: P = f.plot(rgbcolor=(,/4,/)) sage: (P+P).show() The plot (which takes 5 terms of the Fourier series) is given below. Figure : Graph of f(x) = x + and a Fourier series approximation, L =. () This time, let s consider an example of a cosine series. In this case, we take the piecewise constant function f(x) defined on < x < by {, < x <, f(x) =, x <. We see therefore L =. The formula above for the cosine series coefficients gives that 6

7 f(x) + 4 sin ( n) cos( nx n ). The first few partial sums are ( cos S = / + x), ( cos S = / + x) ( cos x),... As before, the more terms in the cosine series we take, the better the approximation is, for < x <. Comparing the picture below with the picture above, note that even with more terms, this approximation is not as good as the previous example. The precise reason for this is rather technical but basically boils down to the following: roughly speaking, the more differentiable the function is, the faster the Fourier series converges (and therefore the better the partial sums of the Fourier series will approximate f(x)). Also, notice that the cosine series approximation S is an even function but f(x) is not (it s only defined from < x < ). For instance, the graph of f(x) and of S are given below: x.5 Figure : Graph of f(x) and a cosine series approximation of f(x). () Finally, let s consider an example of a sine series. In this case, we 7

8 take the piecewise constant function f(x) defined on < x < by the same expression we used in the cosine series example above. Question: Using periodicity and Dirichlet s theorem, find the value that the sine series of f(x) converges to at x =,,. (Ans: f(x) is continuous at, so the Fourier series at x = converges to f() =. f(x) is not continuous at, so at x = the Sine series converges to f(+)+f( ) = f( +)+f( ) = + =. f(x) is not defined at. It s Sine series is periodic with period 6, so at x = the Sine series converges to f odd( )+f odd (+) = + =.) The formula above for the sine series coefficients give that f(x) = The partial sums are S = S = sin ( x) cos (n) cos ( n) + sin( nx n ). sin (/ x) + sin ( x) + sin ( x), sin ( x) 4/,... These partial sums S n, as n, converge to their limit about as fast as those in the previous example. Instead of taking only terms, this time we take 4. Observe from the graph below that the value of the sine series at x = does seem to be approaching, as Dirichlet s Theorem predicts. The graph of f(x) with S 4 is x.5 Figure 4: Graph of f(x) and a sine series approximation of f(x). 8

9 Exercise: Let f(x) = x, < x < and L =. Use SAGE to compute the first terms of the Fourier series, and plot the corresponding partial sum. Next plot the partial sum of the first 5 terms and compare them. References [F] Wikipedia Fourier series article [F] MacTutor Fourier biography: 9